Someone said in a comment that "most Americans didn't want to be part because they were sick of war and wanted to revert back to isolationism". I'm not a sociologist so I won't speak for most Americans living in 1919. However, isolationism wasn't the cause, not in itself at least, but rather one of the key components of isolation: nationalism.

Before I continue, I'd like to clarify something. People sometimes get offended at the mention of nationalism, claiming that it was patriotism. I'd argue that, for the purposes of the study of international relations, nationalism and patriotism have a lot in common. Nationalism needn't be a bad word.

Also, in terms of foreign affairs, the US didn't need to revert to isolationism, because it hadn't stopped being isolated since the turn of the century, even during the Great War.

---

As you rightly said, Wilson, as Theodore Roosevelt before him, had advocated for the formation of an international alliance. But the reason the US didn't in fact enter the League of Nations, was a strong political disagreement rooted on a nationalistic approach. The main advocate against joining under the proposed term, was republican senator Henry Cabot Lodge, who was also the chairman of the Committee on Foreign Relations.

It's important to note that I said he didn't want to join under the proposed terms. Particularly, a condition written in a draft for the Covenant of the League of Nations, specifically article X, which reads

"The Members of the League undertake to respect and preserve as against external aggression the territorial integrity and existing political independence of all Members of the League. In case of any such aggression or in case of any threat or danger of such aggression the Council shall advise upon the means by which this obligation shall be fulfilled."

In terms of what was then understood as international law, the article constituted the undertaking of a binding responsibility, to go to war if and when called upon to do so by the Council and the Permanent Secretariat.

Lodge believed that no foreign power or organization could nor should have the right to be called directly into military mobilization, without going first through the US Congress, because the Congress has to approve a war. In opposition to the Treaty of Versailles, he wrote (rather ironically) fourteen points against it, called the Lodge Reservations.

In a speech given in 1919, Lodge said "(...) The United States is the world's best hope, but if you fetter her in the interests and quarrels of other nations, if you tangle her in the intrigues of Europe, you will destroy her powerful good, and endanger her very existence (...)". (Robert C. Byrd, 1994, The Senate, 1789-1989: Classic Speeches, 1830-1993)

In Toward an Entangling Alliance: American isolationism, internationalism, and Europe, 1901-1950 (1998), Ronald Powaski explains that the rejection of the League wasn't simply a matter of isolation versus internationalism, but rather a matter of being able to control the US' involvement in war, instead of simply handing power over to the League. Furthermore, Thomas Bailey, in Woodrow Wilson and the Great Betrayal, published in 1945 and revised over the decades by him and by other historians, he explains that a division within the democratic side of Congress, regarding whether to accept or not the Treaty of Versailles without amendments, made it easier for the republican side, led by Lodge, to block the signing.

In the end, it came down to Wilson wanting to accept article X, and Lodge refusing. When Wilson was unable to produce a two-thirds majority, the treaty didn't pass. This had two immediate consequences. First, the US didn't join the League of Nations. Second, the US had to sign separate peace agreements with Germany and Austria-Hungary. And in the long term, the Lodge Reservations were victorious once more when, in 1945, the United Nations gave the power of veto to the permanent members of the Security Council.